circuit analysis - What is AC actually?

What is actually alternating current (AC)? And direct current (DC)? I've been reading several books about basic electric circuits analysis and they are contradictory.

1. Some books say that AC is the same as the sinusoidal stationary state. That is, a sine or cosine function (actually, they are the same only shifted horizontally). For instance, \$v(t) = A \sin(\omega t + \phi)\$. Acording to these books, DC is totally stationary (static), that is, the voltage and current are not time-dependant (\$v(t) = k\$, where \$k = \mathrm{const.}\$). Currents have the same form as voltages on each of these cases.


2. According to other books, AC is any current that is changing (that is alternating) his sign over time (\$t\$). With maths notation: \$i(t_1) < 0\$ and \$i(t_2) > 0\$ for some \$t_1\$ and \$t_2\$ where \$t_1 \neq t_2\$. It could have any shape as long as it's alternating his sign.

I'm inclined to think that in practice wins the first option but maybe it's better the etymology of the second option.

Some other cuestions have relation to this:

1. Is AC a voltage that is the sum of a sinusoidal stationary state and a constant voltage? For instance, \$v(t) = A \sin(\omega t + \phi) + k\$.


2. Is AC a shawtooth wave?